Nuprl Lemma : urec-lfp
∀[f:Type ⟶ Type]. ⋂r:{r:Type| (f r) ⊆r r} . r ≡ urec(f) supposing continuous'-monotone{i:l}(T.f T)
Proof
Definitions occuring in Statement :
continuous'-monotone: continuous'-monotone{i:l}(T.F[T]),
urec: urec(F),
ext-eq: A ≡ B,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
apply: f a,
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
member: t ∈ T,
urec: urec(F),
tunion: ⋃x:A.B[x],
pi2: snd(t),
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
continuous'-monotone: continuous'-monotone{i:l}(T.F[T]),
type-monotone: Monotone(T.F[T]),
type-continuous': semi-continuous(λT.F[T]),
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
rev_subtype_rel: A ⊇r B,
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
subtype_urec,
urec_wf,
subtype_rel_wf,
istype-universe,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
subtract-1-ge-0,
continuous'-monotone_wf,
fun_exp0_lemma,
istype-void,
fun_exp_unroll_1,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
fun_exp_wf,
subtract_wf,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
istype-le,
subtype_rel_functionality_wrt_implies,
subtype_rel_set,
subtype_rel_universe1,
subtype_rel_weakening,
ext-eq_inversion,
ext-eq_weakening,
sq_stable__subtype_rel,
subtype_rel_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
independent_pairFormation,
lambdaEquality_alt,
isectElimination,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
dependent_set_memberEquality_alt,
universeIsType,
applyEquality,
equalityTransitivity,
equalitySymmetry,
isectIsType,
setIsType,
instantiate,
universeEquality,
setElimination,
rename,
imageElimination,
productElimination,
sqequalRule,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
voidElimination,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
functionIsType,
unionElimination,
because_Cache,
isect_memberEquality_alt,
voidEquality,
isectEquality,
setEquality,
cumulativity,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[f:Type {}\mrightarrow{} Type]. \mcap{}r:\{r:Type| (f r) \msubseteq{}r r\} . r \mequiv{} urec(f) supposing continuous'-monotone\{i:l\}(T.f T)
Date html generated:
2020_05_20-AM-08_18_22
Last ObjectModification:
2020_02_01-AM-11_13_13
Theory : general
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